Remarks on Physics as Number Theory (original) (raw)
Related papers
An interface between physics and number theory
Journal of Physics: Conference Series, 2011
We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a mathematical route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate inter alia a basis for concluding that the Euler gamma constant γ may be rational.
Number Theory and Physics at the Crossroads
2011
This is the fifth gathering of this series of workshops in the interface of number theory and string theory. The first one was held at the Fields Institute in 2001. The subsequent workshops were all held at BIRS, and this one was the fourth BIRS workshop. The workshop met from May 8th to May 13th for five days. Altogether 40 mathematicians (number theorists and geometers) and physicists (string theorists) converged at BIRS for the five-days' scientific endeavors. About half of the participants were familiar faces, while the other half were new participants. There were 24 one-hour talks, and our typical working day started at 9:30am and ended at 9:30pm. Indeed we worked very hard, and the workshop was a huge success! This fourth workshop was partially dedicated to Don Zagier on the occasion of his completing the first life cycle and reaching the 60 years of age. This series of workshops has been getting very prominent in the community and attracted more applications than the workshop could accommodate. There was overwhelming urge from the participants to organize a five-day workshop in two or three years time at BIRS. We are indeed planning to submit a proposal for the next BIRS workshop in due course.
Toward the Unification of Physics and Number Theory
World Scientific Reports in Advances in Physical Science , 2019
In Part I, we introduce the notion of simplex-integers and show how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. In Part II, we introduce a geometric analogue to the primality test. Our geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound n-simplexes or associated An lattices is discovered, a deep understanding of the error factor of the prime number theorem would be realized – the error factor corresponding to the distribution of the non-trivial zeta zeros. In Part III, we discuss the mysterious link between physics and the Riemann hypothesis. We suggest how quantum gravity and particle physicists might benefit from a simplex-integer based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax. Finally, an appendix provides an overview of the conceptual framework of emergence theory, an approach to unification physics based on the quasicrystalline spin network. 7/26/18 Correction: In Section 3.2 on page 13, it says: Conjecture 2 : if p is composite, let p = ab, where a≠1, b≠1 then at least one of the coefficients is not divisible by p It should say: Conjecture 2 : if p is composite, let p=a b, where a and b are prime and a≠1, b≠1 then at least one of the coefficients is not divisible by p.
The Representation of Numbers in Quantum Mechanics 1
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to +k j−1 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on just the successor for j = 1. This is the only successor defined in the usual axioms of arithmetic.
Non-Diophantine arithmetic as the mathematical foundation for quantum field theory
2022
The problem of infinities in quantum field theory (QRT) is a long standing problem in physics.For solving this problem, different renormalization techniques have been suggested but the problem still persists. Here we suggest another approachto the elimination of infinities in QFT, which is based on non-Diophantine arithmetics - a novel mathematical area that already found useful applications in physics. To achieve this goal, new non-Diophantine arithmetics are constructed and their properties are studied. This allows using these arithmetics for computing integrals describing Feynman diagrams. Although in the conventional QFT these integrals diverge, their non-Diophantine counterparts are convergent and rigorously defined.
The Representation of Numbers in Quantum Mechanics
Algorithmica, 2002
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to +k j−1 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on just the successor for j = 1. This is the only successor defined in the usual axioms of arithmetic.
2019
The concept of waves is very fundamental to classical and modern physics alike, being essential in describing light, sound, and elementary particles, among many other phenomena. In this paper, we show that the wave/particle duality is a phenomenon manifested not only in the physical world and the mathematics that describes it, but also in the simple numbers that form the basic matrix upon which most of our sciences rest. We will also show how this wave-based approach to numbers could be essential to our understanding of the mathematical and physical constants that govern the physical laws as well as the natural elements emerging from them.
Arithmetic Aspects of Atomic Structures
HELVETICA PHYSICA ACTA, 1997
After the initial success to explain the hydrogen atom, one of the early challenges of quantum mechanics was to study larger atoms. The problems encountered in this process were numerous, and the quest for an understanding quickly became a search for simpli ed quantum atomic models that would explain di erent properties of the atom. This study generated a vast theory with rami cations in many areas of physics and specially mathematics, including some which will be reviewed in this presentation: semiclassical asymptotics, eld theories, potential theory, computational issues and analytic number theory.